Completely boundary-free minimum and maximum principles for neutron transport and their least-squares and Galerkin equivalents

Ann.nucL Energy,Vol. 9, pp. 95 to 124,1982 Printedin GreatBritain

0306-4549/82/020095-30$03.00/0 PergamonPressLtd

COMPLETELY BOUNDARY-FREE MINIMUM AND MAXIMUM PRINCIPLES FOR NEUTRON TRANSPORT A N D THEIR LEAST-SQUARES A N D GALERKIN EQUIVALENTS R. T. ACKROYD UKAEA, Northern Division, Central Technical Services, Risley, Cheshire, U.K. (Received 10 January 1981) Abstract--Some minimum and maximum variational principles for even-parity neutron transport are reviewed and the corresponding principles for odd-parity transport are derived by a simple method to show why the essential boundary conditions associated with these maximum principles have to be imposed. The method also shows why both the essential and some of the natural boundary conditions associated with these minimum principles have to be imposed. These imposed boundary conditions for trial functions in the variational principles limit the choice of the finite element used to represent trial functions. The reasons for the boundary conditions imposed on the principles for even- and odd-parity transport point the way to a treatment of composite neutron transport, for which completely boundary-free maximum and minimum principles are derived from a functional identity. In general a trial function is used for each parity in the composite neutron transport, but this can be reduced to one without any boundary conditions having to be imposed. An alternative derivation of the functional identity gives as a by-product Davis complementary principles for composite neutron transport, which use two trial functions satisfying essential boundary conditions. If these two trial functions are replaced by one then both natural and essential boundary conditions have to be imposed. The functional identity is used to re-establish three well-known principles directly, and it shows that the boundary-free maximum principle is equivalent to a generalized least-squares method with weights in the form of operators and no boundary conditions imposed. The least-squares principle uses two positive definite volume integrals so that the divergence theorem can be used to change awkward volume integrals into manageable surface integrals without the need to impose boundary conditions on trial functions. A geometrical interpretation of the boundary-free maximum principle is given using the projection theorem for a Hilbert space with a suitable metric. This path leads to several boundary-free Galerkin equations for both the second-order and first-order forms of the transport equation. I. INTRODUCTION Several one-dimensional p r o b l e m s o n shields (Ackroyd et al., 1978, 1980), reactor cores (Galliara a n d Williams, 1979; Ackroyd et al., 1980) a n d lattice cells; a n d some two-dimensional problems on shields a n d lattice cells (Ackroyd a n d Grenfell, 1979 ; Ziver et al., to be published; Splawski a n d Williams, private c o m m u n i c a t i o n ) have been solved using a m a x i m u m principle for the even-parity one-group n e u t r o n t r a n s p o r t e q u a t i o n (Ackroyd, 1978). This principle has been used to solve a n u m b e r of two-group one-dimensional p r o b l e m s (Ackroyd et al., 1980) a n d some two-group two-dimensional problems (Ziver et al., to be published) for reactor cores a n d shields. The only requirements o n the trial function in the m a x i m u m principle are the essential b o u n d a r y conditions for the continuity of the trial function across the interfaces of the finite elements used in the representation of the trial function a n d the reflection condition for a perfect reflector. F o r some of the above problems (Ziver a n d Q u a h , private c o m m u n i c a t i o n ) a c o m p l e m e n t a r y (minimum) principle for even-parity t r a n s p o r t has been used (Ackroyd, 1978). This principle is not as easy to use as the m a x i m u m principle, because it requires a n additional a n d natural condition o n the gradient of the trial function to be satisfied at interfaces a n d on a perfect reflector. However, the c o m p l e m e n t a r y principle is of practical significance, because using the m e t h o d of bi-variational b o u n d s rigorous local error b o u n d s of practical significance can be o b t a i n e d (Ackroyd a n d Splawski, 1981). F o r example, the captures in the cladding of a fuel element can be calculated to within _+0.29/o (Ackroyd a n d Splawski, 1981) using trial functions admissible in b o t h the m a x i m u m a n d m i n i m u m principles. F o r two a n d three-dimensional p r o b l e m s it would be especially convenient to have a m i n i m u m principle with no more restrictive b o u n d a r y conditions t h a n the m a x i m u m principle. 95

96

R.T. ACKROYD

For the simplest approximation to the Boltzmann equation provided by the neutron diffusion equation the boundary-free variational method of Delves and Hall (1979) could be used in conjunction with global elements. In their method the solution of the diffusion equation makes an associated functional stationary, whereas in the sequel the solution of the Boltzmann equation by boundary-free variational methods is found by either maximizing or minimizing an appropriate functional. Davis (1968) using two trial functions, one for even-parity transport and the other for odd-parity transport, obtained complementary principles for what may be described as mixed parity transport. Local error bounds based on these principles are given by Ackroyd and Splawski (1981). The trial functions have to be continuous at interfaces and satisfy a reflection condition at perfect reflectors. The drawback to this method is that the number of nodal parameters for a finite element calculation is doubled. If one expresses one of the trial functions in terms of the other in a natural way then one is left with a single trial function which has to satisfy two boundary conditions at interfaces. In the sequel, complementary principles are given for mixed parity transport which are completely boundaryfree, i.e. there are neither external boundary conditions nor any internal boundary conditions at interfaces. In general there are two trial functions, one for even-parity transport and the other for odd-parity transport, but either can be expressed in terms of the other to reduce the number of nodal parameters. The complementary principles are derived from a functional identity, the left-hand side of which is given in terms of the trial functions and right-hand side of which is expressed solely in terms of surface and volume sources. By deleting a positive definite functional from the left-hand side, a maximum principle is obtained because this functional vanishes when the trial function is the exact solution. Alternatively the complementary minimum principle can be obtained by adding the positive definite functional to the left-hand side. However, with this new approach there is need only to use either the minimum principle or the maximum principle, and the global error is given directly. To obtain the new principles the complementary principles for odd-parity transport are derived ; and the reasons why the boundary conditions have to be imposed for those principles, and the corresponding principles for even-parity transport, are noted. For each mode of transport (either even or odd-parity) the boundary conditions are imposed to suppress certain terms in two functionals, so that one functional has a maximum value and the other a minimum with the same value. By redefining the functionals it is possible to suppress the unwanted terms by the addition of other functionals. The final result is the functional identity. The boundary-free maximum principle stemming from this identity can be used, by imposing certain restrictions on the trial functions, to generate the well known classical variational principles. The boundary-free maximum principle is equivalent to a generalized least-squares principle which is boundary free. A geometrical interpretation of the boundary-free maximum principle is made with the aid of a suitable Hilbert space; and it leads to boundaryfree Galerkin schemes for both the first- and second-order forms of the Boltzmann equation, which reduce to some conventional Galerkin schemes when boundary conditions are imposed. Thus the well-known equivalence of variational, least squares and Galerkin methods for boundary-tied trial functions is extended to boundary-free trial functions.

2. EVEN- A N D ODD-PARITY TRANSPORT

In solving the Boltzmann equation 12- V~bo(r,12) + a(r)qSo(r,fl) = f o~(r, 12-12')~bo(r,12') d[~'+ S(r, 12) Jn

(1)

for the angular flux q5o (r, 1~) there are some advantages (Ackroyd et al., 1980; Ackroyd, 1978 ; Briggs et al., 1975) to be gained by transforming it into a second-order equation in either the even-parity angular flux ~b+ (r, ~ ) = ½[~bo(r, l~) + ~bo(r, - 12)] or the odd-parity angular flux ~b- (r, 12) = ½[~o(r, 12)-- q~o(r, - 12)]. In place of equation (1) one can use the even-parity transport formulation (Ackroyd, 1978) -- 12- V [G12" VqS~(r, fl)] + C~b0~(r, 12) = S + (r, 12) - 11. V G S - (r, ~ )

(2)

Boundary-free principles: least-squares and Galerkin equivalents

97

or the odd-parity transport formulation* - n . V[-C- 1~. Vq~o(r,l~) ] + G- 1~o = S- - ~ " VC- tS*(r,n)

(3)

where S ÷ and S- are the even- and odd-parity components, respectively, of S. The positive definite operators G- 1 and C are defined directly in terms of a and as, and explicit formulae for G and C - 1are known (Ackroyd, 1978). The scattering and absorption of neutrons in equation (2) is governed by the removal operator C. G is called the leakage operator because the term n - V [ G n . V¢~(r,n)] can be written as div [ ~ G I I • V~- (r, 11)]. Approximate solutions of either equation (2) or (3) can be found using a finite element representation for a trial function by the Rayleigh-Ritz method, or by Galerkin's method. Minimum and maximum complementary principles for the Rayleigh-Ritz method have been derived (Kaplan and Davis, 1967) for both even- and odd-parity transport using the Euler-Lagrange variational method for minimum principles, and by using Friedrich's method for maximum principles. Complementary principles for even-parity transport have been obtained directly (Ackroyd, 1978) and the same method is used in the sequel for odd-parity transport. For the directly-derived maximum principle the trial function has to satisfy the essential or principal internal boundary conditions plus a reflection condition on a perfect reflector. For the directly-derived minimum principle the trial function has to satisfy an additional natural internal boundary condition, which makes it less convenient to use than the maximum principle. By using a trial function for even-parity transport and another trial function for odd-parity transport, Davis (1968) has obtained complementary principles for mixed-parity transport. Each trial function has to satisfy an essential internal boundary condition and surface conditions. The above direct method for obtaining complementary principles for even-and odd-parity transport can be extended to give complementary principles for mixed-parity transport. In general a trial function is used for each of the parity fluxes, but only one is necessary. In this treatment of mixed parity transport, no boundary conditions whatsoever need be imposed on trial functions. These principles are called completely boundary-free to distinguish them from the partially boundary-tied maximum principle and minimum principles reviewed in Section 3. 3. REVIEW OF PARTIALLY BOUNDARY-TIED COMPLEMENTARY PRINCIPLES FOR EVEN-PARITY TRANSPORT

et al.,

Very accurate solutions tk~ (r, ~) of the even-parity transport equation (2) can be obtained (Ackroyd 1978, 1980; Galliara and Williams, 1979; Ackroyd and Grenfell, 1979; Ziver to be published; Ackroyd and Splawski, 1981 ; Splawski and Williams, private communication) by maximizing the functional

et al.,

K(~,) = .Iv [2(~, S ÷ ) + 2 ( n . V4,, CS- ) - ( n " V,~, G~" V~) - (4', C~)] dV

+2fs;a. ..
~n q~21['~"n. df~ dS

(4)

with respect to the nodal parameters used for the finite element representation of the trial function ~(r, l'~) over the system volume V. For brevity the argument of a function is suppressed if it is (r, fl). Here the scalar product denotes ~n u(r, f~)v(r, n ) df), n is the outward normal to the surface of V; and note that the boundaries of physical regions and neutron shadows are outlined by the faces of finite elements. The boundary conditions on ~b~-for the bare surface Ss and the surface Ss with source T are given in part 1. For ~bto be admissible in K(th) it has to satisfy the boundary conditions (i) tk(r, ~) is continuous across the interfaces of finite elements for all directions crossing the ] interface. / (ii) ~b(r,f~) = ~b(r,f~) on a perfect reflector, where fl and fl x are the directions of an incident and reflected beam respectively. * See Section 4.

(5)

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R . T . ACKROYD

The complementary maximum principle is

E(d?) = fv (C-' +fs{fa.

[(B + C)4~- S + + div

IIGS- ], (B + C)d~- S ++ div ~GS )

dV

.,,>o 'FZ'nl[ck+GF~'V~)--T(r'--~)--GS-]2d~+IJ u.,, < o ,D.- n, [~b- G . • V(;b- T

+ G S - ] 2 dO} dS

+f,,{f,,..>ol~'nl[~'+~'V¢-GS-]2dn+f,.~ola'"'[~'-G~'V¢+~S-]'dn}dS + K(q~) t> F(4~, 4~-) >~ K(~b)

(6)

Here, B~b denotes - d i v [ ~ " Gf~" V~b], and F(~b~-,4)o~) is defined to be

r(q~ff, c~ff) = fv

[ ( ~ " V~bg (r, ~), G ~ - V4~ (r, ~ ) ) + (q~- (r, ~), C~b+ (r, ~ ) ) ] d V

+f.... fo .rO,.O.dO S

,7,

For q5to be admissible in the minimum principle E(qS),it must satisfy the boundary conditions (5) and the additional natural continuity condition (iii) G[f~" Vdp(r,f ~ ) - S - (r, f~)] is continuous for all ~ crossing an interface.

(8)

Although the minimum principle is not as convenient as the maximum principle in obtaining approximate solutions it enables the global and local errors of an approximation to be found. The global error of an approximation 4) for q~- in a generalized least-squares sense is (Ackroyd, 1978) F(~ -- q~-, q~-- ~bg) ~< E(~b)- K(q~).

(9)

Precise bounds for local errors are found in the following way using a method of bi-variational bounds. Let V0 m V be a locality of special interest and co a cone of directions ~ at r; then local characteristics of ~b~- of the kinds

fooar,")h(r,a)dadVo f

and

fvf,..Vck~(r,F~)k(r,F~)df~dV,o

with h and k as chosen weighting functions, can be bounded using q5 the approximation for qSg and ~ba an approximate solution for a related auxiliary problem. The span of the bounds is ~/[E(~b) -- K(q~)] [e(~b,) - K(~ba)] .

(10)

4. EQUATIONS OF ODD-PARITY TRANSPORT

F r o m the fundamental equations (Ackroyd, 1978) ~-V¢o +c4,g = s +

(II)

1¢o = s -

(12)

4,6~ = C - ' [ S + - ~ . V ~ b o ]

(13)

a.V4,g+G

and the invertibility of C

the odd-parity equation

-~'V[C-I~'Vdpo]+G

~)ff = S - - ~ . V C - I S +.

(14)

In treating this equation the fact that the operators ~ - V and div ~ are the same is useful in transforming volume

Boundary-free principles: least-squares and Galerkin equivalents

99

integrals into surface integrals, and in ascribing physical meaning to the first term of equation (14). Note that the solutions ~b~ and ~bo are unique when the boundary conditions are of the kinds given below. This follows from the uniqueness of ~b0 = ~b~ +q~o, the solution of the Boltzmann equation (1) for the same physical boundary conditions. The exterior surface boundary conditions for odd-parity transport are obtained in the following way. For the bare surface Sb with n the unit outward normal at r q~o(r,~)=0 i.e. ~b~-(r,~)+~bo(r,~ ) = 0

for~-n<0

for~-n<0 and

~bg(r,~)-~bo(r,~)=0

for~-n>0.

(15a)

Thus C-l[S+-~'~-V(])o]~,-(/~o

=0

forn-n<:}.

(15b)

C-a[S+-~.Vdpo]-~po = O forfl'n>

Similarly for a surface source T(r, fl) in Ss the boundary condition T(r, fl) = ~bo(r,f~)

for l ~ - n < 0

becomes T(r, fl) = ~b~-(r, 1~) + 4)o (r, fl), ~ " n < 0

and

T(r, - - ~ ) = ~ + ( r , ~ ) - ~ b o ( r , f ~ ) , ~ - n > 0.

(16a)

i.e. T(r,~)=~bo+C

l[S+-~-V~bo]

T(r, - - ~ ) = --q~o + C - a [ S + - l ~ ' V ~ b o ] On a perfect reflector

forf~'n<0

]

for ~ ' n > 0~"

Spr ~bo(r,f l ) = ~bo(r,~x) for all ~ ' n = - f l X ' n ~ 0

i.e.

(16b)

(17a)

~b~-(r, fl) = ~b~(r, K~x),~bo (r, fl) = q~o(r, ~x)

where f~ and f~x are the incident and reflected directions of a neutron. Hence S + (r, ~ ) - - S + (r, ~ x ) _ ~ . V~bo(r, ~ ) + fix. V~bo(r, ~x) = 0

(17b)

for all ~ such that l ~ . n = - ~ X ' n 4:0 The boundary condition for an interior surface, or interface, with outward normal n at r is, ~

and ~bo are continuous for all directions f~ which cross the surface

(18a)

i.e. ~bo and C - 1[S+ - l ~ " Vq~o] are continuous for all directions l~ which cross the surface, i.e. for all ~ " [] :¢- 0 (18b) This condition is a direct consequence of the continuity of ~bo across the surface.

5. MINIMUM AND MAXIMUM PRINCIPLES FOR ODD-PARITY TRANSPORT AND THEIR ESSENTIAL BOUNDARY CONDITIONS

5.1. A basic functional

A simple derivation is given below of m i n i m u m and maximum principles for odd-parity transport which do not require the trial functions to satisfy the exterior boundary conditions except on a perfect reflector. For the maximum principle only one boundary condition has to be satisfied at the interfaces of regions, whereas two boundary conditions for interfaces have to be fulfilled for the m i n i m u m principle. The interior boundary condition to be satisfied for the maximum principle is an essential (Friedman, 1956) one for a classical variational treatment. For the m i n i m u m principle one of the interior boundary conditions is essential and the other is natural in the classical sense.

I00

R.T. ACKROYO

The simple derivation is given as it provides the key to obtain maximum and m i n i m u m principles for which the trial functions do not have to satisfy any exterior or interior boundary conditions. In deriving the maximum principle, the basic functional is

F-(~-,I]/-) = ;v [<~'~-V~)-, C- 1~"~"V~/-)--[- <(~-, G- I1~-) ] dV

+LIo,

,,9,

The grounds for choosing this particular functional are the same as the grounds (Ackroyd, 1978) for choosing the basic functional, equation (7), for even-parity transport. An admissible function 4~- for the functional F - has to satisfy two conditions : (a) q~- is a continuous function ofr and ~ almost everywhere within the subregions V~into which V is partitioned by specified surfaces (Fig. 1). These surfaces comprise the boundaries of physical regions, the edges of neutron shadows and the boundaries introduced for computational convenience, e.g. the boundaries of finite elements. Note that within a subregion, cross sections are continuous functions of position. (b) Within each subregion, ~ • Vq~- (r, f~) is almost everywhere a continuous function ofr and ~ .

Ss Sb

{

\, sj ~ Sb

S1

s/-__

---

Fig. 1. Subregions and specified surfaces. The functional F - has the properties : Property 1. I f ~ - (r, ~ ) is admissible then F -(q~ -, q~-) is positive definite because C - t and G - t are positive definite.

(20)

Property 2. Since C - 1 and G - t are linear self-adjoint operators, F-(qS-,O-+~b-) = F-(d?-,O-)+ F-(d?-,~b-) = F-(O- +~b-,dp-) for 4~-, 0 - and ~ - admissible.

(21)

Property 3. Since C - t and G - 1 are linear operators, F-(~b -, ).~k -) = 2F-(~b-, ~b-) = F-(2gb -, ~b-) for 2 a real n u m b e r and 4,-, ~O- admissible.

(22)

5.2. A maximum principle and its essential boundary conditions In demonstrating this principle the trial functions are required to satisfy a boundary condition on interfaces and a reflection condition for a perfect reflector, and for this reason the boundary conditions are called essential here. The boundary conditions on admissible functions ~b- (r, f~) are: on the interface ~b- is continuous for all directions £~ crossing the interface, i.e. I'~- n ~ 0.

(23)

Boundary-flee principles: least-squares and Galerkin equivalents

101

and on the surface of a

perfect reflector ~b-(r, fl) = $ - ( r , l ~ x) for a l l g l - n = - f ~ X ' n # 0 .

(24)

To obtain a maximum principle, note that 0 ~< F - (q~- - ~bo, q~- - ~o) = F - (ff -, ~b-) - 2F(~b -, fro) + F - (~bo, q~o)-

(25)

Also note the fact demonstrated below, that F - (~b-, ~bo) depends only on ~b-, S - , 1"~-V C - 1S + and T(r, l-l). Hence a functional K-(q~-) is defined by K-(~b-) = 2F-(q~-, ~ b o ) - F - ( q ~ - , ~ - ) <~ F-(~bo, ~bo).

(26)

The maximum of K - (~b-) is attained if ~b- = ~bo . To demonstrate this principle it is not necessary to assume that ~b- is an odd function of II. To show that F-(4'-,4,o) =

jv[(¢-,S-)+(a'VO-,C-'S+)]

dV

+ f s , fa-.
(27)

Observe that the fundamental equations (11) and (12) and the divergence theorem give

v[(l~.Vdp-,C-1I'~.V¢o )+(dp-,G-idpo )] dV bU~USpr

Si

where S~is that part of the boundary of the subregion ~ which lies within I/ and n~ is the unit outward normal to Sv On splitting the range ofintegration for f~ into f~" n > 0 and ~ - n < O, and using the boundary conditions given in equations (I 5a) and (16a)

(29)

f~.f4'-(¢~ol~'nl-~'n4'~)d~dS=f~.f,~. ~o[4~-(r,~)-4'-(r,-~)]l~'nlT(r, I S~)d~dS Since

f ~- nO- (r, ~)O~-(r, f~) df~ = f.x ~x" nO-(r, f~x)d,~(r,~ ) df~X the boundary conditions, equations (17a) and (24), give

fsfa~'nc~-~gd~dS=O.

(30)

or

Hence from the definition in equation (19) F-(~b-,q~o) =

fv[(dp-,S-)+(~'Vdp-,C-'S+)] OV

+ffo,.~o[~b-(r'f~)-~b-(r'--~)]lf~'nlT(r'~)d'OdS -fus,fo~.n,~,-e~ffd~dS.

(31)

102

R.T. ACKROYD

The integral over USi vanishes to give the result in equation (27); on account of the boundary conditions, equations (23) and (18a) and the opposite directions of the normals to contiguous subregions. The corresponding maximum principle for even-parity transport has been obtained (Ackroyd, 1978) by defining the functional

F+(~+,~+)=fv[<¢+,Cq,+)]dV+fsbvs.fnln.nlO+q,+dndS

(32)

then K+(~b +) = 2F+(~ +, ~b~)- F+(¢ +, ¢+) ~< F+(¢~ -, ~b~-)

(33)

where F +(~b+,¢~) is independent of ¢~, provided that ~b+ satisfies the boundary conditions: (a) q~+ is continuous across interfaces for all directions ~ crossing the interface.~ (b) ¢ + ( r , ~ ) = ¢ + ( r , ~ x) for all ~ on Spr with ~ ' n = - ~ X . n )

(34)

5.3. Observationson thefurther developmentof variationalprinciples In the further development of variational principles the above boundary conditions on trial functions are waived. The following results are important for the further development. If the boundary conditions, equations (23) and (24), are waived, then V-(¢-'~b°) =

fv [(4~-,s-)+(n.v¢-,c-ls+)] dV

Similarly, if the boundary conditions equation (34) are waived then F+(¢+'q~-)

= fv [(¢ +'S+)+(a'Vd?+'GS-)] dV

--fvs, fnl'~'ni~ +@odD dS-fs,,fn~'nd?+~b o

df~ dS.

(36)

These results in conjunction with two other observations enable one to obtain completely boundary-free maximum and minimum principles for neutron transport. These observations arise in establishing the minimum principle of Section 5.4 for CoIn obtaining the corresponding maximum principle (Ackroyd, 1978) for even-parity transport, which was free of boundary conditions on SbUSsbut dependent on both natural and essential interface boundary conditions, a nonnegative functional was added to K +(q~+). The volume term was of the kind I(¢+,~b-)

= fv ( C - ' [ I ~ -

V~b- + C ¢ + - S + ] , I ~ - V~b- + C ¢ + - S +) dV

(37)

with for this application

¢- = G[S- --1'~.Vq~+].

(38)

For the corresponding minimum principle for odd-parity transport the volume term in the functional to be added to K - ( ¢ - ) is J(~b +, q~- ) = f ( G [ ~ " V~b+ + G - l ~ b - - S - ] , ~ ' V ¢

9v

+ + G - l~b- - - S - ) dV

(39)

Boundary-free principles: least-squares and Galerkin equivalents

103

and for this particular application 4,+ = C- 1[S+ --~-V4,-].

(40)

Having obtained the minimum principle of Section 5.4 for 4,0, complementary boundary-free principles are obtained for mixed-parity neutron transport using the functional I(4, +, 4, -) + J(4, +, q~-) as defined by equations (37) and (39) only. If this combined functional is used with the constraint (38) it reduces to I(4, +, 4)-), and if the constraint (40) is imposed the functional reduces to J(4, +, 4,-). The use of both constraints is in general not feasible, because they would imply that ~b+ and 4, - were within each V~solutions of the fundamental equations (11) and (12). Both 1(4,+, 4,-) and J(4, +, 4,-) are non-negative functionals, because C- 1 and G are positive definite operators. The vanishing of/(4,+, 4,-) implies and is implied by the constraint (40). Similarly the vanishing of J(4,÷,49-) implies and is implied by the constraint (38).

5.4. A minimumprincipleand its boundaryconditions On putting

~+ -~g 0- 4,--4,o J 0 + =

(41)

equations (25) and (26) give for an arbitrary functional d(4, +, 4,-) K-(4,-)+ J(q~+,4, -) =

F-(4,o,4,o)+J(4,+,4,-)-F-(O-,O-).

(42)

The functional J is introduced so that when a suitable exterior boundary functional is added to both sides of equation (42) the l.h.s, is greater than F-(4,o, 4,0)Since 4,~ and 4,0 satisfy equations (11) and (12) and G is self-adjoint,

j(4,+,4,-) = f~ [(fl.VO+,G~.VO+)+(O-,G-IO-)+ 2(O-,Gfl.VO+)] dV. If q~÷ is chosen to satisfy equation (40) then 0+ = -C-'[t~-VO-] and hence using the divergence theorem

J(4,'+,4,-)--F-(O-,O-)=

f

dV

+2fvs, fn O+O-~'n~dfldS+2fJSpr fn O+O-g~'nd~dS

+fnbvs.fn[20+O-~'n-O-zl~'nl]df~ds"

(43)

To proceed to a minimum principle the following exterior boundary surface integrals are added to both sides of equation (42). /b(4, +' 4 , - ) = fS b {fn..>o I"" n[ [~ + -- 4,-]2 d " + oa.° f <0 ["" nl [-4,+ + ~'b-]2 df~} dS / and

dot as. On using the boundary conditions (15a) and (16a)

lb(4~+'~=)+ls(4~+'4'-)=fs, fn {'t~'nl[O+z+O-2]-2~'nO+O-}df~ds" ,~s,

(45)

104

R.T. ACKROYD

Hence

J(~ +,rb-)- F- (O-,O-)+ 6,(~+,~-) + ls(~b÷,rb-)

f. °+°-a'nidads f la.nlO+2dOdS(46) pr

bUSa

Let the following boundary conditions be imposed on ~b+. (a) On each interface ~b+ is continuous for all directions f~ crossing the interface. (b) On the perfect reflector ~b+(r,~) = ~b+(r,~ ~) for all g~ such that g~'n = - ~ ' n

(47)

~ 0

These boundary conditions and the boundary conditions in equations (23) and (24) for tk- imply that both 0 ÷ and 0 - satisfy boundary conditions of the kind (47). Consequently the integrals of 0 +0- over USi and Spr both vanish. Since G and C - 1 are positive definite operators it follows that

J(c~+,q~-)-F-(O-,O-)+ lb(dp+,dp-)+ l~(q~+,~ -) >10 with equality holding if

tk + = q~-

and

~b- = tko.

(48)

Hence equations (42) and (48) give

K-(dp-)+ J(dp+,~-)+ Ib(dp+,dp-)+ l~(dp+,dp-) ~ F-(dpo,dPo ) with equality if

~b+ = q~g and

~b- = ~bo.

(49)

For this principle tk ÷ is given in terms of ~b- by equation (40), ~b- and ~b÷ satisfy the interface conditions (23) and (47a), respectively. The perfect reflector conditions (24) and (47b) have also to be satisfied. The interface boundary conditions for ~b- are essential (in the sense of the calculus of variations) for the maximum principle, equation (26). In the minimum principle, equation (49), the interface boundary condition for tk- is a disguised form of the natural boundary condition for ~b-.

6. COMPLETELY BOUNDARY-FREE COMPLEMENTARY PRINCIPLES FOR COMPOSITE NEUTRON TRANSPORT

6.1. Key remarks The key to obtain completely boundary-free complementary principles for neutron transport is provided by observing the reasons for imposing the essential boundary conditions to obtain complementary principles for even and odd neutron transport. In deriving the maximum principle K-(~b-) above the essential boundary conditions are; (a) continuity of ~b- at interfaces for all directions I'~ crossing the interface and (b) the reflection condition q~-(r, ~ ) = ~b-(r, f~x) for all ~ - n = 0, where ~ ' n = - ~ X ' n . Previously it was found that the essential boundary conditions for the maximum principle K +(~b+) were of the kinds (a) and (b) with ~b- replaced by ~ +. The essential boundary conditions of the kind (a) arise from the need to make both

e-=fv f foOgO-~'nidmdS n ~ ° ~ + a ' nand' dP+=fv f ~ ds, S s , vanish. Similarly, the boundary conditions of the kind (b) are required to make both

pr

pr

Boundary-free principles: least-squares and Galerkin equivalents

105

vanish. Observe that on account of the interface boundary conditions of the kind (18a) for ¢~" and ¢o

where it is understood that the values assigned to ¢+ and ¢ - on S~ are those yielded by the ¢+ and ¢ - defined within the subregion V~.In equation (50) ¢+ and ¢ - are arbitrary and 0 ÷ = ¢+ - ¢ ~ , 0- = ¢ - - ¢ o . Similarly, since ~b~ and ~bo satisfy a reflection condition of the kind in equation (17a),

pr

pr

These results suggest that instead of making the integrals P-,P+, Q- and Q+ vanish to obtain maximum principles for even- and odd-parity transport, one should look for a functional combining both even- and oddparity transport in such a way that the unknown integral

fv fa~'n,O+O-d~dS+fs fn~'nO+O-dfldS=P?(say) Si

(52,

pr

does not arise when a maximum principle is sought. The procedure is to define the functionals K +(~b+), K - (¢-), L(~b+, ~b-) and M(¢ +, ¢ - ) as in Section 6.2, then one shows that K +(¢ +)+ K - ( ¢ - ) + L(¢ +, q~-)+ M(~b+, ~b-)+ F+(O+,0+)+ F-(O-, 0-)+ 2P?= F+(dpff,¢~)+ F-(dpo,C~o) and also that

(53) l(~b+, ¢ - ) + J(¢ +, ¢ - ) + lb(~b+, ¢ - ) + Is(¢ +, q~-) =

F+(O+,O+)+F-(O-, 0 - ) +

2P?.

(54)

These results lead to the functional identity

K+(dp+)+K-(qS-)+ l(¢ +,dp-)+J(¢ +,dp-)+L(d?+,c~-)+M(¢ +,¢-)+ tb(dp+,¢-)+ ls(c~+,¢-)

the basis for the maximum principle. Having found the functional identity it can be verified directly as in Appendix A.

6.2. A functional identity Define for arbitrary admissible functions ~ + and ~ K+(~b +) = 2fv [ < ¢ + , S + > + < a ' V ¢ + , G S - > ]

+2fs. fa..
dV-F+(¢+,~b +)

- ~ ) ] I~" nl T(r' ~ d ~ dS

2fv

dV-F-(¢-,¢

+2fs. fo..
M(q~+,~b-) = - - 2 ~

)

- n)] In',,I r(r, a ) d a dS

= -2fv , fo L~-~b+a-n

(56)

(57)

dn dS

(58)

d . dS

(59)

pr

then from equations (35) and (36) with equations (50) and (51)

K+(c~+)+K-(c~-)+ F+(¢+,¢ +)+F-(c~-,¢-)+ L(¢ +,c~-)+M(dp+,¢-)+ 2p? = 2[F+(~b+,~b~)+F-(~b

,~bo)]

106

R.T. ACKROYD

and hence

K+(@+)+K-(dp-)+L(c~+,~b-)+M(dp+,4,-)+F+(O+,O+)+F-(O-,O-)+2P?=F +(~b~-,4,~-)+ F - (qSo, ~bo). (60) Since 4'ff and 4,o satisfy the fundamental equations (11) and (12), expanding the scalar products for 1(4' +, 4,-) and J(4, +, ~b-) gives I(4,+,q~-)+J(4,+,4,-) =

F+(O+,O+)+F-(O-,O-)+2p? -fs ~t,s.{fa..>o ' ~ ' n l

[0+-0-]2

df~+ [.)a..
'II'"'[O++O-]2dt2} ds"

(61)

The definitions (44) with the boundary conditions (15a) and (16a) give

lb(dP+'4'-)+ls(dP+'dP-)=fs~vs,{fa.,>o]~'nl[O+-O-]2d~+fa.n
I(4'+,4'-)+ S(q~+,dp-)+lb(@+,~-)+ I,(~ +,dp-) = F+(O+,O+)+F-(O-,O-)+ 2P?

(62)

Hence from equation (60)

K +(4,+) + K- (4,-)+ J(~b+, ~b-)+ l(q5+, 4,-)+ L(4,+, 4,-)+ M(4, +, 4,-) + Ib(4,+ ,4,-)+ I~(4,+,C~-) = F ÷(4~ff, 4,$) + F - (4~o, ~bo)

(63)

Since 4, ÷ = 0 = 4,- are admissible and I(0, 0), J(0, 0) and Is(0, 0) are the only non-zero terms on the l.h.s, of equation (63)

F +(4,if,4,+)+F-(qgo,4,0)

=fv[+<°s'S->]dW+2fsf, .

.,<0111""[T(r'f~)2 dia=2c~(say)"

(64)

Hencethe fundamentalidentity,equation(55).

6.3. A maximumprincipleand its 9eneralizedleast-squarescounterpart To obtain a boundary-free maximum principle from the fundamental identity, equation (55), two non-negative functionals are defined which give measures of the mis-match of trial functions at interfaces and on perfect reflectors. On the interface Sic~Sjof the subregions Viand V,.,Fig. 2, there are allowed at r jumps 4' + (r + 0, l l ) - 4' +(r - 0 , ~ ) and 4 ' - ( r + 0 , ~ ) - 4 ' - ( r - 0 , ~ ) in the trial functions. A measure of these jumps for the system is the interface functional If(qS+,~b-) = ; u

f n [~" nl {w + (r, ~)[4, +(r + 0, ~ ) - 4, + ( r - 0, ~)12 (Sif~Sj)

+ w- (r, fl)[4, - (r + 0, ll)-- ~b- (r - 0, ll)] 2} dO dS where w ± are positive weights and n is normal to the interface interfaces

Sf~Sj. Since q~ff and

/f((~O-, ~bo) = O.

On the perfect reflector Spr , Fig. 2, there are allowed at r the jumps 4, +(r, ll)-- 4, +(r, ~ ~) ~b (r, Kl)-tb-(r, fl ~)

for all ~ - n = --KIx- n # 0,

(65) 4,0 are continuous across (66)

Boundary-free principles: least-squares and Galerkin equivalents

107

\~ n

$pr

x "--------- ~ __..._.~.

4'(r- 0, fl) # th(r+ 0, fl) in general iZ f~.n = -flX.n f'~-O

r

si

~.

s,

s1

4'(r, fl) # tp(r,1~~) in general ~

Jump in 4'(r, l)) at the interface Si c~ S i of subregions V~and ~ bounded by surfaces and Sj, respectively,

Si

l

Jump at r on perfect reflector Spr between incident beam 4'(r, f~) and reflected beam 4'(r, flx).

Fig. 2. Jumps in 4' for interfaces and a perfect reflector. where fix is the reflected direction to f l and n is normal to Spr. A measure of these jumps is the perfect reflector functional Ipr(¢~ + , ¢ ~ - ) =

_fs.,fo

,~'nl

{w + (r,

+(r,")-

2

+ w - (r, [i) [ $ - (r, n ) - ~b- (r, f~x)] 2} dr) dS

(67)

and lpr(~b~-,$o) = 0.

(68)

The weights w(r, fl) are prescribed according to the requirements of a particular problem. To proceed to a maximum principle define the non-negative functional W+ -(~b+, ~b-) =

l(d?+,dp-)+J(d?+,d~-)+Ib($+,4a-)+Is(c~+,(a-)+lr(d?+,$-)+lp,((a+,d?-)

(69)

K+(dp+)+K-(d?-)+L(4a+,d?-)+M(4a+,c~-)-lf(c~+,dp-)--lpr(c~+,$-)

(70)

and the functional K+-(t~+, $ -) = then K + - ( 4 , + , 4 , - ) + W+-(O+, ~ - ) = F +(t#~, ~b~-)+ F-(~bg, ~bo)

=fv[(C-aS+'S+)+(GS-'S-)]dV+2f

fa JS s

In" nl T(r'[~)2 dr) dS .n
2or.

(71)

In general, K + -(~b +, ~b-) ~< F +(~b~, ~b~-)+ F-(~b o, ~bo) = 2~

(72)

holds for arbitrary admissible functions 4~+ and q~-, which are not necessarily even and odd functions of fl, respectively. If ~b+ = ~bg and ~ - = 4~o, it follows from the definitions of the components of W + - that equality is attained in the maximum principle, equation (72). If equality is attained in equation (72) then, (Appendix B), ~b÷

108

R. T. ACKROYD

and +- satisfy the parity transport equations and boundary conditions. Also, 4 = fC9+(r,Q+4+(r,

-Ql

-W+4-(r,W--4-(r,

satisfies the first-order Boltzmann equation and its boundary conditions. Furthermore $,$’ and 4- are unique, (Appendix C). Also, 4’ and 4 - are even and odd functions of Q respectively, (Appendix D). Hence equality holds in equation (72) if 4’ = 40’ and & =&. The maximum principle, equation (72), has been given for arbitrary trial functions 4’ and do-. Only one trial function need be used if one of the substitutions & = G[S--42*Vb’]

(a)

r$+ = C-‘[SC -fi*V&]

(b) 1

(73)

is made. Both substitutions are impractical, because they would imply that both 4’ and o- satisfied the fundamental equations (1) and (2) for the parity angular fluxes. Suppose the substitution (73a) is made, then J(b’, 4) vanishes and W+,4-)

=

ssY n

{4~V[G~~V~+]+C~+-S++S-k~VGS-} x C-‘{-~~V[GQ~V~+]+C~+-S++S2*VGS-}

dR dV.

(74)

If C-’ were a positive weighting function instead of a positive definite operator then by equation (2) the volume term in W + - {4’, G[S- -a *Vd’]} would be of the classical least-squares kind (Friedman, 1956). One may regard W’ -(4’, d-) 2 0, in which 4’ and 4- can be independent, as a generalized least-squares method in which weights are replaced by positive definite operators and two trial functions can be used instead of one. Furthermore, in this generalization the trial functions do not have to satisfy any boundary conditions, whereas in the usual least-squares method boundary conditions are imposed on the trial function. In maximizing K+ -(4+,4-) one minimizes W+ -(f$+, f#-); consequently the completely boundary-free maximum principle K+ -(d+, 4-) < 2~ and the generalized least-squares principle W’ -(4’, CJ-) > 0 are equivalent. Let the residuals E+(T,a) and E-(r,Q be defined by Q.V$+ +G-‘&-Sn.v4-

+cl#J+ -s+

= c+ = E-

for arbitrary trial functions 4’ and d-, then the volume component of W’-(d’,4-)

s

[(G~+,e+)+(c-‘E-,&-)I

is

dV > 0.

V

The surface components of Wf -(c$+, c#-) are non-negative also. Hence a measure of the global error made by the maximum principle can be taken to be W’-((#J+,l#J-)/2a. 6.4. A minimum principle Instead of subtracting W”(#J’, c#-) from FC(4i, 4:) +F-(&,&) (72), it can be added to give the alternative minimum principle E+-@+,+-)

= K+-(4+,4-)+2W+-(4+,4-)

7. ALTERNATlVE

DERIVATIONS

to give the maximum principle, equation > F+(~o’,~o’)+F-(~~,&).

OF THE FUNCTIONAL

(75)

INVARIANT

7.1. Two functional equalities

The functional identity, equation (SS),has been verified directly in Appendix A. Another way of establishing the identity is as follows, and on the way Davis’s (1968) complementary principles are established as a special case. The first step is the setting up of two functional equalities, which can be used in two ways. These equalities can be

Boundary-free principles: least-squares and Galerkin equivalents

109

turned into inequalities by suppressing positive and negative unknown quantities, and by eliminating internal boundary terms by imposing essential boundary conditions. This way leads to Davis's principles. The other way is to sum the equalities and add a known set of functionals to eliminate all unknowns. This way leads back to the functional invariant. From equation (36) and the boundary conditions for ~b+, ~bff on interfaces and on a perfect reflector

F+(4~+'4~g)=fv[
..
Similarly from equation (35) -.
= 2~

(76)

F-(q~o, dpff)-F+(dp~, dp~) = fv E(C- 'S +, S +) -- (GS-, S- ) -- 2(q~o(r, --n), So)] dV -2f~fo, ..
(say)

(77)

where So = S + + S - . Since F+(O+, 0 +) = F+(~ + , ~ - ) + F +(~b~, ~b~)- 2F +(~b+, ~b~)

equations (36) and (56) give

J USi

J Spr

Si

JSpr J[~

Similarly

A

l

Hence from equations (76) and (77)

dUSi d ~

JSpr d f l

= O[4~o(r,-~)] (78) 7.2. The complementary principles of Davis and their essential boundary conditions

In the special case where ~b+ and q~- are made continuous across interfaces and satisfy the reflection condition for a perfect reflector, equation (78) reduces to Davis's (1968) complementary principles K+(~b+)--~ ~< Q[~b0(r, - ~ ) ] ~< ~ - K - ( ~ b - ) .

(79)

ll0

R.T. ACKROYD

A simple derivation of these principles has been given by Splawski and Williams (private communication) for onedimensional systems.

7.3. Return to the functional identity The span of the bounds in the above special case is 2 ~ - K +(~b+ ) - K - ( 4 , - ) . In the boundary-free general case equation (78) gives 2c~--K+@ +) - K - @ - ) =

F+(O+,O+)+F-(O-,O-)--2fvs,__ __fnU" n,@+4,ff

df~ dS

--2fsfn a-=(4,+•o +4,-4,g) dn dS.

(80)

tar

The surface integrals on the r.h.s, can be rewritten using equation (52) as L(~ +, ~b-) + M@+, 4,-) + 2P? and in conj unction with equation (62), equation (80) gives the fundamental identity (equation (55)). By reversing the above steps it can be seen that the complementary principles, equation (79), can be derived from the fundamental identity (equation 55)).

8. A P P L I C A T I O N S

OF

THE

FUNDAMENTAL

IDENTITY

8.1. General remarks The fundamental identity (equation (55)) has been used to establish variational principles. Davis's complementary principles provide one example. Three well-known principles are derived in Section 8.2. + + + Complementary principles which bound either F (tko, q~0) or F - @ o , ~bo) are given in Section 8.3 as bounds of these kinds are useful in determining local error bounds with some precision (Ackroyd and Splawski, 1981). Local error bounds can be used to establish benchmarks of high precision where analytical solutions are unavailable. The boundary-free principles permit the use of non-conforming elements.

8.2. The fundamental identity as a generator of variational principles Example (i): to show that K +(~b+) ~< F ÷(~b~, ~b~-),if ÷ is continuous at each interface for all directions crossing the interface ) (a) ~b 4~+ (b) satisfies the reflection condition 4,+(r, l l ) = 4~+(r,~ x) for all ~ such that l l . n = l ~tx" n ~ 0 on a perfect reflector.

(81)

-

Put 4~- = 4~o, then because ~bo satisfies continuity and reflection conditions of the kinds (a) and (b) L@ +, ~bo) = M@ +, ~bo) = 0. Also from equation (27) and the definition (57) g - @ o ) = F - @ o , 4,o). The fundamental equations (11) and (12) give

i(4, +, 4,o) + J(4, +, 4,0) = ~v [<°+' c o + > + <¢t. v0 +, G ~ . V0 + >3 dV. The choice 4,- = 4,o implies 0- = 0, and hence from equation (45)

lb(&,~o)+1,(4,+,4,o)=fsbvs, falft'n'O÷2dt~ds" Substituting the above results in the form of equation (63) of the fundamental identity gives the maximum principle (Ackroyd, 1978) K+(~b +) = F+(4,0,4,+)--F+(4, + --4,~-,4,+--4,~-) ~< F+(4,~-, 4,g).

(82)

Boundary-free principles: least-squares and Galerkin equivalents

1t 1

Example (ii): if ~b- satisfies boundary conditions of the kind in (81) then putting ~b+ = ~b~ in the fundamental identity leads to K - (~b-) = F - (~bo, ~bo ) - F - (~b- - ~bo, ~b- - ~bo) ~< F - (~Po, ~o)

(83)

the maximum principle given in Section 5.2. Example (iii): if both ~b÷ and qg- = GES- - ~ " Vq9+] satisfy boundary conditions of the kind in (81), then E+(~b +) >~ F + ( q ~ , q ~ )

(84)

where* E+(qb +) = ;v
+~.VGS-)

dV

+

+ b

{Io

In.nlEd)-+Gl~.Vdp+-GS-]

2 dQ+

-n>O

r

In.nl[~p+-Gn.Vc~++GS-]

dfl.n
+ K + (q5*).

t

2 dQ dS (85)

With the above choice for $ - , J(~b +, q~-) vanishes and l(~b+, qS-) becomes the volume term in the expression for E +(q5+). Also the surface functionals I,(4 +, ~b-) and lb(q9 +, q~-) become the surface integrals of E + (~b+) over S, and Sb, respectively. Furthermore, L(~b+, ~b-) and M(~b+, ~b-) both vanish, because both ~b+ and ~b- satisfy boundary conditions of the kind in (81). Thus the fundamental identity (equation (63)) reduces to g-(~b-)+E+(q9 ÷) = f+(~bg,~b~)+f-(~bo,qSo). Since qS- is also admissible for equation (83) E+(~b +) = F+(¢~,~b~')+ F-(~b - - t k o , qb--~bo) >I F+(~b~, ~b~).

8.3. Upper and lower bounds for F ÷( qb~ , d)~ ) and F - ( d?o, q~o ) Here upper and lower bounds are obtained for F + (the, ~b~-),and similar bounds can be obtained for F - (tho, ~bo ). If ~b+ satisfies the boundary conditions in (81) then a lower bound for F + (q~, ~b~) is provided by equation (82), and if $ - complies with the same kind of boundary conditions then equation (83) gives a lower bound for F - (qbo, q5o ). Since the sum of F+(tk~, q~) and F-(~bo, q~o) is given by equation (63) K+(~b +) ~< F+(~bff,q~) ~< K+(ck+)+J(ck+,ck-)+l(dp+,dp-)+l~(~k+,c~-)+Ib(¢~+,4:-),

(86)

the terms M(~b+, qS-) and L(q~+, ~b-) vanishing on account of the boundary conditions in (81). In the special case where the two trial functions are replaced effectively by one on putting then

ok- = G E S - - f ~ . V ~ +]

).

K+($+) ~< F+(~b~_,q~. ) ~< E+(~b+)j,F

(87)

because J(~b +, qg-) vanishes, I(~b+, q~-) becomes the volume term in E +(q~+), and I~(~b+, ~b-) + Ib(~b+, ~b-) becomes the surface term on E÷(q~+). The inequality in expression (87) also follows directly from the inequalities in equations (82) and (84). The span of the bounds (equation (86) is 2 , t - K +(q5+ ) - K - (q~-) because equation (64) holds. The trial functions can be determined therefore by minimizing the span of the bounds instead of maximizing K +(th +) and K - (~b-) separately. * Apart from the notation this is the same result as equation (6).

112

R.T. ACKROYD

9. A GEOMETRICAL INTERPRETATION OF M A X I M U M PRINCIPLES A N D GALERKIN METHODS AS A BY-PRODUCT

9.1. General remarks The way in which maximum principles provide approximate solutions can be described geometrically as the minimization of the length of a vector in a Hilbert space with a suitable metric. This vector is the difference between the vector representing the exact solution and the vector corresponding to the trial function. The geometrical interpretation of the optimum trial function gives the Galerkin equations equivalent to those obtainable from the maximum principle for the optimum trial function. The Galerkin equations are obtained in effect by the method of orthogonal projection (Friedman, 1956). This equivalence of the Galerkin method and maximum principles is well-known for self-adjoint operators when the trial functions satisfy the essential boundary conditions of the problem. This equivalence is illustrated for the K ÷ principle, with its essential boundary conditions, and then extended to the completely boundary-free K ÷ - principle. Several Galerkin schemes are obtained as special cases of the general scheme. Thus the boundary-free K ÷ - principle, or its generalized least-squares equivalent, can be regarded not only as the generator of the classical variational principles, which hold for trial functions constrained by the essential boundary conditions, but also as a generator of Galerkin methods. Some of the Galerkin methods given are boundary-tied, and others are boundary-free. Some of these methods apply to the second-order transport equations, and others treat the firstorder Boltzmann equation.

9.2. The K + principle and its Galerkin equivalent Let U ~ [u(r, fl)] be an arbitrary vector in a Hilbert space with scalar product*

U'V = fvE+] dV+ fsb~s,fn In'nluv dn ds = v'U"

(88)

This expression is permissible for a scalar product because G and C are self-adjoint operators. Since G and C are positive definite, the length [[Ull = Ux/UTU = x / ~ is positive definite, i.e. the space has a positive definite metric. Represent the solution by the vector @~- ~ [~b~-], and an approximation to it by M

• +*--,[~b +]

where

q~+= ~ amdp+~ (say))

M

i.e.

*+=

2 amO~+

,,,: 1 and

O~+~-~[~b~ +]

m=l

[

l

(89)

The trial function 4 + is chosen to be admissible in the K ÷ principle. It is assumed that the ~ + are linearly independent, i.e. ~ + lies in the linear M space generated by the O=+. ' The K + (~b+) maximum principle stems from the result K+(~b +) + t +(~b~- -~b +, ~b~--q~+) = t + (q~-, ~b~)

(90)

which can be interpreted geometrically in Fig. 3 as follows. Since {D+2 = F+(~b +, ~b~-) and (~---cD+) 2 = f + ( q ~ - - - ~ + , ~ - - - ~ b +) the function q~+ which maximizes K+(q9 +) minimizes the distance I I O ~ - ® + l l of ~ + from ~ ¢ ; and for a minimum M

X amO+m"tilt+ = ¢l[~ff" tI~/+.

(91)

ra=l

* This choice of scalar product is suggested by the fact that U 2 = F+(u, u), the basic functional for the K ÷ principle. See McConnell (1951) for the relationship between scalar products and variational functionals.

Boundary-free principles: least-squares and Galerkin equivalents

113

roximatJonvector~:~ s~lult|-o¢i - vk:teorr~ecl°r

If ( ~ - Oo)2 is minimized ( ® - ~0)" O~ = 0 ( O - ~o)" ~ = 0 Fig. 3. Hilbert space interpretations of max K +(~b+) ~
( @ ~--~- * ++)" ) ' *@: + =00} " (@g

(92)

The scalar product

0 = (@~ -- @ +)" aP+ = ;v [ + ] d V +;sbvs. fn Ifl" nj ¢+(¢~- - ¢+) d " dS

(93)

can be transformed, on using the even-parity equation (2), the boundary conditions for Cm + and ~b~ and the divergence theorem, to,

0 = fv (c~+' - n ' V G [ n ' V ¢ + ] + C ¢ +

-S+ + n . V G S - )

dV

+ GS-) dil t dS

a form of Galerkin approximation for the even-parity equation. If for example the trial function is chosen to satisfy all the boundary conditions then equation (94) reduces to the simplest Galerkin approximation

o=~fn¢+~{--n'VGEn'V¢+]+Cck+--S++n'VaS-}dndV for the even-parity equation.

(95)

114

R.T. ACKROYD

The odd-parity transport equation can be treated in the same way to obtain Galerkin approximations. For both the even- and odd-parity Galerkin equations the weights tk~+ and ~b~ in the volume term are the basis functions for ~b÷ and ~b-, respectively.

9.3. The K +- maximumprincipleand its direct Galerkinequivalents For the interpretation of the K ÷ - principle consider a Hilbert space which associates with the four functions ui, uu, Uiu and uiv of r and f l the vector U ~ {ui, uu, Uui, Uiv}.The scalar product is defined to be* U - V = f v [(Gum'/)ili) + ( C - lUiv,/)iv))] dV

f In" .1 {w+(r,n)[u,(r+ 0, n)- ui('-O, a)3 [/)i(r+O,a)-/)'(r-O'a)3 (SinS j)

+ W- (r, ~ ) [uH(r + O, ~ ) -- u.(r -- O. ~ ) ] [/)u(r + O, ~ ) --/)ii(r -- O, ~ ) ] } d ~ dS +

fs fn

I ~ ' n l {w + (r,,-q)[ui(r,-q)--ui(r,~)] [vi(r,f~)-/)i(r,f~O]

pr

+ W- (r, ,Q) [uii(r, .Q) -- uii(r, .Qx)]. [/)ii(r, ~"]l)--/)ii(r, $~x)]} df~ dS where

[l'n=-~X-n

on

Spr.

(96)

Since G and C - 1 are positive definite operators, U 2 is positive definite. A region of the space associated with the variational principle K ÷ - is that where vectors U are defined in terms of a single function u(r, 1~) by Ui = U +,

Uii : U -

Uiii = ~'~" V u + + G -

(97) l u - , Uiv = ~[~" V u -

+Cu

+

where u ÷ and u - are the even and odd components of u, respectively. In this region U 2 becomes U2 =

;v [ ( G ( ~ ' V u + +G-lu-)'~'Vu+ +G-lu-)+(C-l(~'Vu- +Cu+)'fl'Vu- + C u r ) ]

dV

+fsbus.{fn..>ol~'nl(u+-u-)2d~+fn..
+fsfnlll'nl{w+[u+(r,~)--u+(r, flx)]Z+w-[u-(r,f~)--u(r,~X)]2}df~dS

(98)

pr

and the way is open to interpret the K ÷ - maximum principle geometrically as follows. The solution vector ~ 0 has two elementary properties; viz., (i) the sum of the first two components gives q~0, and the sum of the last two components gives the extraneous distributed source S, and (ii)*2 =

fv [(GS-'S-)+(C-1S+'S+)]

dV+2 1

t

dS~ J n . n < O

(equation (71)). * Note U" V = V- U because G and C- t are self-adjoint.

I " ' n . T 2 df~ dSgivesther.h.s, side ofthe identity

Boundary-free principles: least-squares and Galerkin equivalents

115

For the out-of-kilter vector a ' - - O0 ,--, [,~ + - - , ~ , ~ - - ~o, f~" V(4, + - ~ ) + G - '(4,- - ~o), f~" V(4,- - 4,o) + C(,k + - ~ ) ] it follows, from the parity equations and the boundary conditions for ~b~- and ~bo, that the contributions to ( 0 - 0 0 ) 2 from the volume V and the surfaces SbUSs, U(SinSs) and Sp, are ,/(~b+,q~-)+I(~b+,q~-), Ib(~b+, ~b-) + I~(~b+, q~-), If(~b+, ~ - ) and Ip,(ff +, qb-), respectively. Hence ( 0 - 0 0 ) 2 = W ÷ -(c,b +, ~ - )

(99)

and the identity (equation (71)) becomes K + -(~+,~b-)+(O-Oo)

2 = 0 2.

(100)

Thus maximizing K + -(q~ +, q~-) is equivalent to choosing the approximation vector • so that the distance in the function space of • from the solution vector • o is minimized. Let the approximation • be of the kind u • = ~ a,,,O., m=l

with the +

-

ore,--, [4,,~, q~., ~ - v¢, 2 + a - 'qV,, ~ - v~,7, + c,/,.+ ]

(101)

a linearly independent set of vectors constructed from a suitable set of functions ~b,.(r,~). Since the maximizing of K + -(q~ +, q~-) with M

~b+= ~ amq~+

and

M

~*

~b-= ~

amdP~,

m=l

(102)

m=l

minimizes (O-- 00) 2, then for (l = 1. . . . . M)

E amOm'01 =

Clio'01'

m=l

i.e.

(O0--O)'O t = 0

and

(Oo-O)'O

(103)

= 0.

These orthogonality conditions lead in the same way as in the ca~e of the K ÷ principle to the Galerkin type of approximation. O," (00 - O) = 0 = f v {( ' ~ ? ' - a " V[Gf~ • V¢ + ] + C4, + - S + + f~" VGS- ) + ( ~,-, - a " V[C - ' a - V~ - ] + G - l q ~ - - S - + I ' I - V C - 1S+)} dV

+

~[

III'nlEO~(dP++GfI'Vd?+-GS-)+dPF(ck-+C-III'Vdf-C-1S+)]

d~

~b { . ~ ] f l . n > 0

ID"nI[dP~-(dP+-GI~'Vdp++GS-)+dP[-(dP--C-ID"Vdp-+C-IS+)]d~} dS

+

a..<0

+

,. { f a . . > o

II~'nI[dP~-(d?++G~'Vdp+-GS--T(r'-I~))

+dpi-(dp-+C-'~'Vdp--C-'S ++ T(r, --~))] +fn-.
df~

I~'nl[dP:(dP+--Gl'~'VqS++GS-- r(r'f~))

* It is desirable, but not necessary, that the qb~ and qS~ be even and odd functions of fl, respectively.

116

R.T. ACKROYD + 4h- (4)- -- C - 1ft. V ¢ - + C - 1S + - T(r, fl))] df~} dS

+ [ a - n l [w + (4)t - 4)t+ (r, ax)) (4) + - 4) + (r, l'~x)) + w - (4)7 - Ct- (r, a*)) (4)- - 4)- (r, fix))] } dfl

+fus, f. n'n'{4);-(¢-+Ga'v4) +-6s-)+4);-(4)++c-'n'v4)--c-'s+)} dn ds + fu(s,,~sj)fa Ifl'nl {w+ [¢+(r + 0, f ~ ) - ¢ ~ + ( r - 0 , ~ ) ] [4)+(r + 0 , 1 2 ) - 4) + ( r - 0 , f~)] +w-[4)t-(r+0,fl)-4)t-(r-0,~)]

[4)-(r + 0 , ~ ) - 4 ) - ( r - 0 , f l ) ] }

d n dS(I = 1..... M)

(104)

which is free of boundary conditions. Note that the weights ¢l+ and 4)1- in the volume term of the Galerkin equations are the bases for 4) ÷ and 4)-, respectively. If all the boundary conditions are satisfied, and the 4)~+ are taken to be independent of the ¢,-, then and

;vf"¢:{-n'VEGa'V4)+]+C4)+-S++n (105)

fvfa4)i-{-n.V[C-~n.vc-]+G-'4)--S-+n.VC-'S+}dndV=O the simplest Galerkin approximations for the even- and odd-parity equations respectively. If one chooses an even-parity approximation (equation (102)), which satisfies for arbitrary am the continuity of 4) + across interfaces and the reflection condition on SD,, and takes

4); = 4)o/al

with

4)~-,= 0

for

m> 1

(106)

the approximation (equation (104)) reduces to the Galerkin scheme, equation (95).

9.4. A second-order Galerkin approximation using a boundary-free even-parity trial function only This Galerkin approximation is derived from the K ÷ (4)+, 4)-) principle by choosing 4)- = G[S- - - n " V¢ +]

(107)

as follows. Put M

0+=

~ a,~4)~+

(108)

m=l

where the 4)~+ are chosen even functions of fl, and define the odd function M

4)- = ~ am4)~ +GS-

(109)

ra=l

where 4):, = - G n . v4) +.

(I t0)

Let M

•

= tD'+ ~

a=(l) m

m=l

where Om ~ [4)+, -- G n " V4) +, O, n " V4); + C4).+ ]

• ' ~ [0, GS-, S-, ~ " VGS-].

(111)

Boundary-free principles: least-squares and Galerkin equivalents

117

m=l approximation for ®o the solution vector of known magnitude.

*2=~v{(GS-,S-)+(C-'S+,S+)}dV+2I I

I..nlT2df~dS

d S . dfl.n
,oo-o,,

a.O.

is minimized by ( ~ o - 0 ' ) ' 0 t = ~, am~,.'~t

(1 = 1..... M)

m=l

i.e. (~o-q~).(~-qP') = 0 Fig. 4. Hilbert space interpretation of K + -(~b+, ~b-) with q~- = G[S- -l'~. Vq~+]. The ~,,+ are chosen so that the 0 . , are linearly independent. The geometrical interpretation of O is shown in Fig. 4. The maximization of K + -(~+, {p-) is shown by the identity (equation (100)) to imply the choosing of • to minimize the distance of • from 0o. For a minimum the a,, have to satisfy M

amO,.'O!

= (0o--0')'01

(l = 1 . . . . . M ) .

(112)

m=l

Since the vectors • m are linearly independent, and hence there is a unique minimum with the orthogonality property (0o-0)-0

t= 0

(l = 1. . . . . M)

(113)

implying

(O--Oo)- ( O - - O ' ) = 0.

(114)

The fundamental equations (1 !) and (12), (107), (110) and the self-adjointness of C - 1 enables one to write the volume term of equation (113) as

- [ "
9:2

- E

118

R.T. ACKROYD

The contributions of ¢~ and ~bo to the surface terms of equation (113) can be removed on using the boundary conditions for ¢~- and ¢o. Consequently equation (113) becomes the Galerkin approximation 0 = ; v (¢'+ - C-'f~" VGEf*" V¢'+]' - f ~ ' W E S a ' V ¢ + ] + C¢+ - S + + f ~ - V C S - > dV

÷

-

+fu

(S~mSg)

-

f. In''l{w+[4?]*-[¢+]+-+w-rGn'v¢?]~-[Gn.v¢+-Gs-]*-) dads

+ fs fn I~'nl{w+[¢[]x[¢+]x+w-[Gl'~'v¢~-]x[G~'v¢+-Gs-]x}

dOds

(115)

gtr

where [u]_+ and [u] x denote u(r + 0, f l ) - u(r- 0, II) and u(r, ~ ) - u(r, El*), respectively. In this example the weights $f~ -C-ll}'V[GI'~" Vq~z+] are not in general the same as the basis functions ~b~+ for q~+. If linear elements are used then ¢ + can be chosen to be continuous across interfaces in one dimension, but not in two or three dimensions. On the other hand ¢ - is discontinuous for all three dimensions. The volume term in the Galerkin approximation simplifies for linear elements to

f (¢?,C¢ + - s

+ +f~"

VGS-)

dV

and the Galerkin equations have the classical form.

9.5. A first-order Galerkin approximation based upon a boundary-freeeven-parity trialfunction only Let equations (107) and (110) hold; then the volume term of equation (115) becomes, with ¢ = ¢ + + ~b- and S = S+ +S-,

f (4'? +C-'[n'V¢i~],n'V¢+C¢ ++ G - ' C -

-S)

dV.

On noting the parities of ~b+ and q~-, the definitions (Ackroyd, 1978) C¢+ = aqS+ - f n ' a~+(r'l~'fl')q~+(r'fl') dt2' G

=

a~b - -

dnr,a~-(r, El. 1~')¢ -(r, l-~') d O '

give ;

C ¢ + + G - 1 ¢ - = a~b- |

p as(r, f l • fl)¢(r, f l )r dO r.

(116)

Thus the volume term of equation (115) becomes (117)

Boundary-free principles: least-squares and Galerkin equivalents

119

The surface terms of equation (115) can be recast, on noting the parities of ~b+ and ¢ - , so that the Galerkin equations become

--2fs f"b . . < o " ' n q ~ t ~ b d f ~ d S - 2f d sj.a . . < o . - n C z ( ¢ - T )

dDdS

+ w- [~bz-(r, f~) - q~z-(r, ~x)] [¢ - (r, ~) - ~b- (r, ~x)] } dtq dS

+fu

(SickS j)

f. If~"nl {w+ [¢?(r + °' f~)- ¢'+(r- °' f~)] [~+(r + °'")- ~ +(r- °' f~)]

+ w - l ¢ { ( r + 0 , f l ) - q S z - ( r - 0 , f ~ ) ] [~b-(r+0,~)-¢-(r-0,1"l)]} dr1 dS

(118)

9.6. Anotherform of a boundary-freefirst-orderGalerkinscheme Let ¢+ and q~- be the even- and odd-parity components, respectively, of a boundary-free trial function ¢. Define the even- and odd-parity residuals by R + = ~-V~b- + C ~ + - S + = - [ ~ ' V ( 4 ' o - ¢ - ) + C ( ¢ ~ - - ¢ + ) ] R-

] f

I'~-V¢+ + G - ' q ~ - - S - = - [ l ' ~ ' V ( t k ~ - ~ b + ) + G - ' ( t k o - t k - ) ] . J

(119)

Thus • o - ~ - - ' [¢~ -¢+,4~o - ¢ - , - R

, -R+].

(120)

Let Iv be the volume term of 0," ( ~ o - ~) in equation (103), then using the scalar product equation (96) and the self-adjointness of G and C - 1

Iv -fv

[ ( R - , Cz- +G~'V~bz+> + (R+,~bz+ + C-1~'V¢~-73 dV.

Since ~ = ¢+ +~b-, ~b+ is an even function ofl'l and ~b- an odd function of D;

Iv --fv (R+ +R-'~bz+Gl~'VCz+ + C -

lfl'Vtkf ) dV.

Since S+ +S- = S

and equation (116) holds

R ++R- = ~'V¢+C~b ++G-ltk--S = ~ ' V ¢ + a q ~ - ;~ as(r, fl . ~ )t ¢ ( r , ~ )p d D ' - S .

(121)

Thus R ÷ + R - is the residual for the first-order Boltzmann equation when ~bis taken as an approximate solution. Having regard to the parities of ~ ÷ and ¢ - one can show in the same way as equation (116) that Gl'l" V~bz+ + c - ' a . v¢,- =

O

a.

+ f/(r, du

a. a')a' •

a') d~'

= ~bz (say)

(122)

where f is given (Ackroyd, 1978) in terms of the scattering cross sections. Hence

Iv = --fa (~b,+ ~b,,a" Vth +tr~b- fu, as(r,f~" lT)¢(r,a') d f ~ ' - S )

dV.

120

R.T. ACKROYD

The surface terms of ~ " (O 0 - O ) are simplified readily on using the boundary conditions for 4,g and 4,0Thus the orthogonality condition, equation (103), reduces to the Galerkin equations

+ w - [4,7 (r, n ) - 4,~-(r, a~)] [4'- (r, n ) - 4,-(r, ~x)]} d n dS

+fo,s,~s,,flsa'nl{w+[4,7(r+o,a)-4,?(r-o,a)][4,+(r+o,a)-4,+(r-o,n)] +w-[4,i-(r+O,fl)-4,i-(r-O,l~)] [4,-(r + 0 , f ~ ) - 4 , - ( r - 0 , ~ ) ] } d ~ dS (1 = 1. . . . . M).

(123)

If the substitution (equation (110)) is made then these Galerkin equations reduce to the Galerkin equation (118). 10. DISCUSSION The functional identity, equation (55), has been identified by examining the reasons why essential boundary conditions are invoked in the usual treatment of extremum principles for second-order forms of the mo0oenergetic transport equation. Having been identified it can be derived readily as in Appendix A. The identity can then be used to establish a boundary-free maximum principle; from which the various maximum and minimum principles in current use can be established by imposing the essential boundary conditions of classical theory, and also in one case (the E(4,) principle) the natural boundary condition. The boundary-free maximum principle is equivalent to a boundary-free generalized least-squares principle, which differs from the usual least-squares approach by using two positive definite volume integrals instead of one to gauge the out-of-kilter of the approximate solution from the exact solution. The boundary-free maximum principle has been interpreted geometrically by representing the approximate and exact solutions as vectors in a Hilbert space with a metric suggested by the functional form of the maximum principle. The role of the maximum principle is to minimize the vector representing the out-of-kilter of the approximate solution from the exact solution. This observation leads by way of the projection theorem to a boundary-free Galerkin method for which a number of variants are displayed. Galerkin schemes appropriate to the second- and first-order forms of the transport equation are obtained. For some of these variants boundary conditions are imposed to reduce them to recognizable schemes. The boundary-free schemes in Sections 9.5 and 9.6 employ weights which are not the same as the bases used to represent the trial functions. The above general treatment provides a unified approach to boundary-free variational, least-squares and Galerkin methods for the Boitzmann equation. In developing a new technique (Ackroyd and Splawski, 1981), such as applying the method of bi-variational bounds to provide upper and lower bounds for a characteristic such as capture in a small region, a variational method is natural; but for other problems a Galerkin scheme may be more appropriate. For some problems there may be practical advantages to be gained by using boundary-free trial functions in problems for which the essential boundary conditions for the trial function can be satisfied easily. For explicit methods of solving the transport equation (such as the SN method), in which the solution is obtained by sweeping through the mesh in the direction of neutron streaming, there are some advantages CReed and Hill, 1973) to be gained in allowing the angular flux to be discontinuous across mesh boundaries. Some advantages may also hold for the implicit methods usually employed to solve directly the variational or Galerkin equations of a problem. For the same angular representation for the flux the maximum principle ensures that discontinuous trial functions are at least as good as continuous ones, but the discontinuous trial functions require significantly more unknowns per node of the spatial mesh. However, the discontinuous trial function does permit a high-order angular representation for the flux where it matters and a low-order representation elsewhere.

Boundary-free principles: least-squares and Galerkin equivalents

121

REFERENCES

Ackroyd R. T. (1978) Ann. nucl. Energy 5, 75-94. Ackroyd R. T. and Grenfell D. (1979) Ann. nucl. Energy 6, 563-577. Ackroyd R. T. and Splawski A. B. (1981) Submitted to Ann. nacl. Energy. Ackroyd R. T., Galliara J. and Williams M. M. R. (1979) Proc. MA FELA P Ill Conf. Mathematics of Finite Elements, Brunel University, 1978 (Edited by Whiteman J. R.). Academic Press, New York. Ackroyd R. T., Ziver A. K. and Goddard A. J. H. (1980) Ann. nucl. Energy 7, 335-349. Briggs L. L., Miller W. F. and Lewis E. E. (1975) Nucl. Sci. Engng 72, 205-217. Case K. M. and Zweifel P. F. (1967) Linear Transport Theory. Addison-Wesley, New York. Davis J. (1968) Nucl. Sci. Engng 65, 127-146. Delves L. M. and Hall C. A. (1979) J. lnst. Maths Applic. 223-234. Friedman B. (1956) Principles and Techniques of Applied Mathematics. Wiley, New York. Galliara J. and Williams M. M. R. (1979) Ann. nucl. Energy 6, 205-223. Kaplan S. and Davis J. (1967) Nucl. Sci. En41ng64, 166-176. Lancaster P. (1969) Theory of Matrices. Academic Press, New York. McConnell A. J. (1951) Proc. R. Irish Acad. 54A, 263-90. Mitchell A. R. and Wait R. (1977) The Finite Element Method in Partial Differential Equations. Wiley, New York. Reed W. H. and Hill T. R. (1973) Proc. Conf. Mathematical Models and Computational Techniques for Analysis of Nuclear Systems. Am. nucl. Soc. 1, 10-31. Splawski A. B. and Williams M. M. R. Private communication. Synge J. (1957) The Hypercircle in Mathematical Physics. Cambridge Univ. Press, Massachusetts. Vulikh B. Z. (1963) Introduction to Functional Analysis for Scientists and Technologists. Pergamon Press, Oxford. Ziver A. K. and Quah C. S. Private communication. Ziver A. K., Goddard A. J. H. and Ackroyd R. T. To be published.

APPENDIX

A

Direct verification of the fundamental identity (55) Expanding the scalar products in equations (37) and (39) for 1(~ +, ~b-) and J(~ +, ~b-), and using the definitions (56) and (57) for K+(q~+) and K-(q~-) with the definitions (32) and (19) for F+(tp+,~b +) and F-(~b-,O-) gives

l(dp+,dp-)+J(dp+,dp-)+K+(dp+)+K-(dp-)

=fv[(C-'S+,S+)+(GS-,S->]dV-fs~o,.I.l"'nl[~+'+,-2ldndS +2f (

3s .Jfl.n
[~b+fr,f~)+~+(r,-l~)+~-(r,l~)-~b-fr,-~)] la'nl T(r,a) df~ dS

+2fvfnfL'V(4~÷,O-)d"dV. Hence, using the definitions (44) for lb(o +, O-) + 1,(O +, q~-),

l(d) +,c#-)+ J(c~+,d)-)+ K+(O +)+ K-(d)-)+ lb(O+,d?-)+ l~(c~+,qS-)

=fv[(C-'S+,S+)+(GS-,S-)]dV+2fsfu

ll'~"nlT(r,fl)2d[~ dS .a
-2fsbr:s.fnl'l'nck+dP-dDdX+2fvfnf~'V(dP+,dP-)df2dV. Thus from the definitions(58)and (59)and the divergencetheorem

~(d~+~c~-)+ J(fb+~-)+ K+(c~+)+ K-(dp-)+lb~ +~dp-)+l`(d~+~c~-)+L(dp+~c#-)+M(~b+~dp-)

=~[adV+2~ ~

Ill" nl T(r,n) 2 d n ds-

d su . / [ i . a < 0

APPENDIX

B

If the admissible functions $ ÷ and $ - make W+-(¢+ ~b- ) =

l(¢+,c~-)+J(dp+,~-)+lb(dp+,dp-)+ls(dp+,dp-)+If(dp+,dp-)+lpr(dp+,dp -)

vanish, then almost everywhere t~+ and ~b- satisfy the parity transport equations and boundary conditions. Furthermore,

122

R . T . ACKROYD

"almost everywhere ~b = ½[~b+(r, f t ) + ~b÷(r, - f t ) + gb-(r, f t ) - ~ b - ( r , - f t ) ] satisfies the first-order Boltzmann equation and its boundary conditions. Since the functional components of W ÷ - are all nonnegative, the vanishing of W ÷ -(~b ÷, q~-) implies (a)

l(~b+,~b -) = 0 = J(~b+, ff - )

(b)

t~(~+,~ -) = 0 = / ~ ( ~ + , ~ - )

(C)

[f(~+, ~b-) = 0 =/pr(~b ÷, ¢-).

The pair (a) show that almost everywhere within each subregion the parity equations are satisfied ft-Vq~-(r, ft)+C~b+(r, ft) = S+(r, ft) ~'V~b+(r, f t ) + G - ~q~-(r, ll) = S-(r, ft). Let ~b(r,ft) = ½[~b + (r, ft) + ~b+(r, - ft) + ~b- (r, 11) - ~b-(r, - ft)] then ft'Vq~(r, ft)+½C[~b+(r, ft)+q~+(r, - f t ) ] +½G-'E~b-(r, f t ) - ~ b - ( r , - f ~ ) ] = S+(r, f t ) + S - ( r , ll) = S(r, ft) because S ÷ and S - are even and odd functions of ft, respectively. The definitions of C and G - 1 give C[4~ + (r, ft) + ~b+ (r, - ft)] = a(r) [~b + (r, ft) + ~b+ (r, - 11)]

- f

a~(r, ft. ft') [q~ +(r, ft') + ~ + (r, - ft')] d~'

G - 1[~b - (r, ft) - gb-(r, - ft)] = a(r) [q~- (r, f t ) - ~ - (r, - ft)] -

1 as-(r, f t dll

ft') [~b-(r,

ft') - q~-(r,

- ft')]

d~'

where

as+ (r, a- ft') = ½[a,(r, ft. fl') +

=,(r,

- ft"

a')]

as-(r, ~" ~') = ½[a~(r, ~- ~') - =~(r, - f t " ft')] are the even- and odd-parity components of the differential scattering cross section a~(r, ft" ft'). Hence ½CI-(~ + (r, ft) + ~ + (r, - ft)] + ½G-' [~ - (r, ft) - ff - (r, - ft)] = =(r)q~(r, 12)

- ~ l " as(r, ft" ft')~(r, ft') dD' .]tl -½~

cq(r, - f t - f t ' ) ~ r ,

- f l ' ) dD'

1' =

o~r)~(r, ft) --

j~, a,(r, fl" ft')~b(r, ft') d~'.

Thus within each subregion gb satisfies the first-order Boitzmann equation I I . V~b(r, ft) + a(r, ft) - I " as(r, ft" ft')~b(r, ~ ' ) d13' = S(r, ft) do

(B.1)

The pair (b) show that the parity boundary conditions are satisfied for S b U S s and give almost everywhere ~b(r, ft) = 0

on Sb

for l l . n < 0

(B.2)

~b(r, gZ) = T(r, ft)

on S,

gl-n < 0

(B.3)

Boundary-free principles: least-squares and Galerkin equivalents

123

as the consequence of

and

~+=~b-

onSb

and

~b++~b-=0

o n S b with

~b+-~b-=T(r,-fl)

onSs

forfl.n>0,)

~b++q~-=T(r,~)

onSs

for~.n<0.

J

From the first member of (c); ~b+ (r, fl) and ~ - (r, fl) are continuous across each interface for all directions f~ crossing the interface. Hence, i.e. ~b(r,f~) is continuous on each interface for all directions crossing the interface.

(B.4)

(B.5)

The second member of (c) gives for all f l ' n = --flX'n # 0 ~ + ( r , a ) = ~b+(r,a~), ~ b - ( r , ~ ) = q~-(r,~ ~) i.e.

~b(r,I'~) = gb(r,l~).

(B.6)

Thus equations (B.1)-(B.3) and (B.5), (B.6) show that ~bsatisfies the first order Boltzmann equation and its boundary conditions almost everywhere.

APPENDIX If W +

C

-(tp +, tp-) = 0, then (i) (ii)

~b+ and ~ - are the unique solution pair of the parity transport equations and boundary conditions ~b = ½{~ + (r, fl) + tp +(r, --fl) + ~b- (r,~)-- ~-(r, - fl)} is the unique solution of the first-order Boltzmann equation and its boundary conditions.

(i) In Appendix B the vanishing of W + -(q~+, ~b-) was shown to imply that ~b+ and t~- satisfy the fundamental parity equations and their boundary conditions. Suppose the solution of this problem is not unique, then the parity components X + and X - (say) of the difference between two solutions would satisfy the source-free problem with the parity equations ~'VX-+CX

+= 0

~'VX + +G-1X - = 0

(C.I)

and the boundary conditions X++X X+--X

- =O,~'n 0)

X+(r, fi) = X+(r,~X) ~

X-(r,a) = X - ( r , a x) )

on S b U S ,

for all ~ - n = - ~ X ' n ¢= 0 on Spr

are continuous at an interface for all directions ~ crossing the interface.

(C.2)

(C.3)

X ÷ and X-

(C.4)

For the source-free problem specified by (C.1)-(C.4) the terms of the fundamental identity simplify as follows : I ( X ÷, X - ) = 0 = J ( X +, X - ) on account of (C.1) 0 = I b ( X + , X - ) = l s ( X + , X - ) = L ( X +, X - ) = M ( X + , X

- ) on account of equations (C.2)-(C.4).

Also, since there are no sources, K+(X+)=-F+(X+,X

+)

and

K-(X-)=-F-(X-,X-).

Hence the fundamental identity (55) becomes F+(X+,X+)+ F-(X-,X

- ) = O=¢, X + = 0 = X -

almost everywhere. Hence t~ + and ~ - are unique. (ii) In Appendix B it was shown that the vanishing of W + -(q~+,~b-) implied that ~b satisfied the first-order Boltzmann equation and its boundary conditions. If the solution of this problem were not unique the even-parity component X + and the odd-parity component X - of the difference X of two solutions would satisfy equations (C.1)-(C.4) of(i), and the fundamental identity would give X + = 0 = X - . Hence X = 0 and ~b is unique.

124

R . T . ACKROYD APPENDIX

D

If W + - ( 4 +, 4 - ) = 0, then almost everywhere 4 + and 4 - are even and odd functions of 11, respectively.

Let ¢(r, 11"I) = ½[ 4 ÷ (r, 11) - 4 +(r, - 11) + ¢ -(r, 11) + 4 - ( r , - 11)]

The vanishing of W + - ( 4 +, 4 - ) implies, as shown in Appendix B,

11"V4- + C 4 + = S + 12-V4 ÷ +G-14~ - = S almost everywhere; whence I I . V~b+ ½G-1 [ 4 - (r, fl) + 4 - ( r , - fl)] + ½C[4 ÷ (r, f i t - 4 +(r, - fl)] = 0 because S + and S - are even and odd functions of El, respectively. The definitions of G - 1 and C give ½G- 1[ 4 - ( r , fl) + 4-(r, -[Z)] + ½C[4 +(r, l Z ) - 4 +(r, -11)]

= ,r(r)a/- f . o . ( r , . ' . ' ) ~ ( r , ~ ' ) d~' + f . ,rs(r, - . ' 1 1 ' ) 4 ( r , - . ' ) d :

Hence

~ . v ¢ +,r¢ = o, the first-order Boltzmann equation for a purely absorbing system. According to Appendix B the vanishing of W ÷ - ( 4 +, 4 - ) implies that 4 " and 4 - satisfy the boundary conditions (B.4)-(B.6). Thus ~k=0

onSbUS~

forll.n<0

¢ is continuous on an interface for all directions f l crossing the interface ¢(r,11)=~(r,11 ~) f o r a l 1 1 1 " n = - f Z x ' n ~ 0

onset'.

Thus ~b satisfies the first-order Boltzmann equation and its boundary conditions for a system without sources. A solution is ~k = 0, and this is unique (Appendix C). Hence - 4 + ( r , ~ ) + 4 + ( r , - - 1 1 ) = 4-(r,1~)+ 4 - ( r , --11), but the l.h.s, is an odd function o f f l and the r.h.s, is an even function o f ~ . Hence 4+(r, f l ) = ¢ + ( r , - - 1 1 )

and

4-(r, fl)=-4-(r,-11).